Classical electromagnetism by Jerrold Franklin

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21), we have (x, y, z) —> (r, 0, #) point transformation and in eq. 22), it is (r, 6, 4>) —»(x, y, z) point transformation. 5 Relationships between space variables (x, y, z), (r, 6, and (p, , z). 4 35 SPHERICAL COORDINATES (r, e, ) The components of vector A = (Ax, Ay, Az) and A = (Ar, Ae, A^) are related by substituting eq. 23) into eq. 2) and collecting terms. 26) A^ = — A* sin + Ay cos 0 In matrix form, the (Ax, Ay, Az) -> (Ar, Ae, A$) vector transformation is performed according to = A.

Each answer can be used once, more than once, or not at all. 10 A wedge is described by z = 0, 30° < < 60°. Which of the following is incorrect: (a) (b) (c) (d) (e) The wedge lies in the x — y plane. 10b. 3 (a) If V = xz — xy + yz, express V in cylindrical coordinates, (b) If U = x2 + 2>>2 + 3z2, express U in spherical coordinates. 8 Prove the following: (a) ax • a p = cos ^> a x a 0 = -sin^ 3 ^ - 3 ^ = sin <£ 3 y • 3 0 = COS 0 (b) a x • a r = sin 6 cos a^. 10 (a) Express the vector field H = xy2zax + x2yzay in cylindrical and spherical coordinates, (b) In both cylindrical and spherical coordinates, determine H at (3, —4, 5).

9) into eq. 2) and collecting terms. Thus A = (Ax cos + Ay sin )ap + (~AX sin + Ay cos 0 ) a 0 + Azaz (2. 14) A, or directly from eqs. 10). 15) A. An alternative way of obtaining eq. 15) is using the dot product. 16) The derivation of this is left as an exercise. 4 SPHERICAL COORDINATES (r, 0, (/>) The spherical coordinate system is most appropriate when dealing with problems having a degree of spherical symmetry. 4. 4 SPHERICAL COORDINATES (r, e, 33 point P or the radius of a sphere centered at the origin and passing through P; 6 (called the colatitude) is the angle between the z-axis and the position vector of P; and 4> is measured from the x-axis (the same azimuthal angle in cylindrical coordinates).

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